let f be a differentiable function such that f(1) = π/2 and f(x) = 3 arctan(x² - 3x + 2). what is the value…

let f be a differentiable function such that f(1) = π/2 and f(x) = 3 arctan(x² - 3x + 2). what is the value of f(3)? a 0.243 b 1.328 c 2.899 d 3.321

let f be a differentiable function such that f(1) = π/2 and f(x) = 3 arctan(x² - 3x + 2). what is the value of f(3)? a 0.243 b 1.328 c 2.899 d 3.321

Answer

Explanation:

Step1: Use the fundamental theorem of calculus

By the fundamental theorem of calculus, $f(3)-f(1)=\int_{1}^{3}f^{\prime}(x)dx$. So $f(3)=f(1)+\int_{1}^{3}f^{\prime}(x)dx$.

Step2: Substitute the given values

We know $f(1)=\frac{\pi}{2}$ and $f^{\prime}(x) = 3\arctan(x^{2}-3x + 2)$. Then $f(3)=\frac{\pi}{2}+\int_{1}^{3}3\arctan(x^{2}-3x + 2)dx$.

Step3: Evaluate the integral numerically

Using a calculator or software (such as a graph - ing calculator with integral - evaluation capabilities) to find $\int_{1}^{3}3\arctan(x^{2}-3x + 2)dx\approx1.328$. And $\frac{\pi}{2}\approx1.571$. Then $f(3)\approx1.571 + 1.328=2.899$.

Answer:

C. 2.899